Conditional Probability

#Definition

Conditional probability is the likelihood of an event occurring given that another event has already occurred or is known to be true. In prediction markets, conditional probability describes how the probability of one outcome changes based on information about related outcomes—expressed mathematically as P(A|B), read as "the probability of A given B."

Understanding conditional probability is essential for trading correlated markets, interpreting market movements, and updating beliefs as new information arrives.

#Why It Matters in Prediction Markets

Conditional probability is the mathematical foundation for rational belief updating in prediction markets. When news breaks or related markets move, conditional probability governs how other market prices should adjust.

Correlated markets: Many prediction markets are interconnected. The probability that a candidate wins a general election is conditional on whether they win their primary. A market on economic recession affects markets on interest rate decisions. Traders who understand these relationships identify mispricings others miss.

Information processing: Every piece of news changes conditional probabilities. A candidate's scandal affects not just their election market but all downstream markets—vice presidential selection, policy outcomes, party control. Sophisticated traders update across all affected markets simultaneously.

Bayesian reasoning: Prediction markets reward Bayesian thinking—systematically updating probabilities based on evidence. Conditional probability is the mechanism for this updating, distinguishing informed traders from those who anchor on outdated beliefs.

#How It Works

#The Basic Formula

Conditional probability is calculated as:

P(A|B) = P(A and B) / P(B)

Where:

• P(A|B) = probability of A given B has occurred
• P(A and B) = probability that both A and B occur
• P(B) = probability that B occurs

#Visualizing Conditional Outcomes

#Python: Calculating Conditionals from Data

If you have historical data (e.g., "How often did the Primary winner go on to win the General?"), you can calculate conditional probabilities directly.

import pandas as pd

# Sample Data: Political Candidates
data = {
    'Candidate': ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J'],
    'Won_Primary': [True, True, False, True, False, True, False, True, False, True],
    'Won_General': [True, False, False, True, False, False, False, False, False, True]
}

df = pd.DataFrame(data)

# Calculate P(General | Primary)
# Formula: P(General and Primary) / P(Primary)

primary_winners = df[df['Won_Primary'] == True]
general_winners_from_primary = primary_winners[primary_winners['Won_General'] == True]

p_primary = len(primary_winners) / len(df)
p_general_given_primary = len(general_winners_from_primary) / len(primary_winners)

print(f"Probability of winning Primary: {p_primary*100}%")
print(f"Conditional Probability (General | Primary): {p_general_given_primary*100}%")

#Bayes' Theorem

For updating beliefs based on new evidence, Bayes' theorem extends conditional probability:

P(A|B) = [P(B|A) × P(A)] / P(B)

Where:

• P(A) = prior probability of A (before new information)
• P(B|A) = likelihood of observing B if A is true
• P(A|B) = posterior probability of A (after observing B)

#Numerical Example in Prediction Markets

Consider two related binary markets:

• Market 1: "Candidate X wins primary" — trading at $0.70 (70%)
• Market 2: "Candidate X wins general election" — trading at $0.45 (45%)

What's the implied probability that X wins the general election given they win the primary?

Using conditional probability:

P(General | Primary) = P(General and Primary) / P(Primary)

If X must win the primary to win the general:

P(General and Primary) = P(General) = 0.45
P(General | Primary) = 0.45 / 0.70 = 0.643 (64.3%)

The market implies a 64.3% chance of winning the general election conditional on winning the primary.

#Updating on News

A trader holds a position in an election market at $0.50. Polling data releases showing the candidate ahead by 5 points. Historical analysis suggests:

• P(5-point poll lead | candidate wins) = 0.75
• P(5-point poll lead | candidate loses) = 0.25

Using Bayes' theorem to update:

P(Win | Poll) = [0.75 × 0.50] / [(0.75 × 0.50) + (0.25 × 0.50)]
P(Win | Poll) = 0.375 / 0.50 = 0.75

The posterior probability rises from 50% to 75%. If the market only moves to $0.60, the trader sees an opportunity.

#Conditional Markets in Practice

Modern platforms like Polymarket use Conditional Tokens to create explicit "If X, then Y" markets.

• Example 1: "Will Candidate X win IF nominated?"

  • This isolates candidate strength from nomination chances.
  • If Candidate X is not nominated, the market resolves to "N/A" (trades are typically reversed or resolve to 0.5/1.0 depending on specific rules), meaning you only bet on the general election performance conditional on getting there.

• Example 2: "Will Bitcoin hit $100k IF ETF is approved by Date Z?"

  • Traders can express views specifically on the impact of the ETF approval, filtering out the scenario where the ETF is rejected.

These markets allow for pure expression of conditional belief: $P(Outcome | Condition)$.

#Examples

Primary-to-general correlation: A trader notices that a presidential primary market trades at $0.80 while the general election market for the same candidate trades at$0.35. The implied conditional probability of winning the general given winning the primary is only 43.75% ($0.35/$0.80). If the trader believes this conditional probability should be higher based on historical data, they buy general election shares as underpriced.

Policy outcome chains: Markets exist for "Fed raises rates in March" ($0.60) and "Recession by year-end" ($0.30). A trader calculates the implied conditional probability of recession given a rate hike and compares it to economic models. If the market underestimates how much rate hikes increase recession probability, opportunities exist.

Event sequencing: In a legal proceeding, markets cover "Defendant found liable" and "Damages exceed $1 billion." The damage amount is conditional on liability—if liability probability drops, damage market prices should adjust proportionally. Traders who recognize these dependencies faster than the market profit.

Sports conditional markets: "Team advances to finals" trades at $0.40. "Team wins championship" trades at$0.25. The implied probability of winning the championship given reaching the finals is 62.5%. A trader who believes this team matches up poorly against likely finals opponents might short the championship market relative to the finals market.

#Risks and Common Mistakes

Ignoring base rates: Traders often overweight new information and underweight prior probabilities. A dramatic news event may feel significant but barely moves true probabilities. Always anchor updates to reasonable base rates.

Confusing correlation with causation: Two events may be correlated without one causing the other. Market movements in correlated events don't necessarily indicate direct causal relationships—a common source may drive both.

Independence assumptions: Treating events as independent when they're correlated leads to probability errors. "Candidate wins State A" and "Candidate wins State B" are not independent—they share common factors like national mood and candidate quality.

Neglecting conditional structure: Some traders buy "X wins general election" without considering it's conditional on winning the primary. If primary odds drop, general election odds should drop proportionally—missing this connection means missing trading opportunities.

Overconfident updating: Bayes' theorem requires accurate estimates of likelihoods P(B|A). Overconfident likelihood estimates produce overconfident posterior probabilities. Uncertainty about the model should temper belief updates.

Conjunction fallacy: The probability of A AND B cannot exceed the probability of A alone or B alone. Yet traders sometimes price conjunctions higher than their components—a mathematical impossibility that creates arbitrage opportunities.

#Practical Tips for Traders

• Map market dependencies before trading—identify which markets are conditionally related and how information flows between them

• Calculate implied conditionals between related markets to spot inconsistencies; prices implying impossible conditional probabilities signal opportunities

• Update incrementally using Bayes' theorem rather than making dramatic probability jumps on single pieces of news

• Track your base rates for different event types; historical frequencies provide anchors for conditional updates

• Consider multiple information pathways—the same news may affect several markets differently based on conditional relationships

• Build spreadsheets modeling conditional relationships between your positions; this reveals correlated exposures and hedging opportunities

• Be skeptical of extreme conditional probabilities implied by market prices; very high or very low conditionals often indicate mispricing or misunderstanding

#Related Terms

• Expected Value (EV)
• Conditional Tokens
• Information Aggregation
• Arbitrage
• Binary Market
• Categorical Market
• Kelly Criterion

#FAQ

#What is conditional probability in simple terms?

Conditional probability is how likely something is to happen when you already know something else is true. In prediction markets, it's the probability of one outcome given that a related outcome occurs. For example, "What's the chance a candidate wins the general election, given they've already won the primary?"

#How does conditional probability differ from regular probability?

Regular (unconditional) probability considers an event in isolation—P(A) is the chance A happens regardless of other information. Conditional probability—P(A|B)—incorporates known information, calculating A's probability given that B is true. Conditional probabilities are always relative to some specified condition.

#Why do prediction market traders need to understand conditional probability?

Many prediction markets are interconnected. Understanding conditional probability helps traders: (1) identify mispricings between related markets, (2) update beliefs correctly when news arrives, (3) recognize when market prices imply impossible probability relationships, and (4) manage correlated positions effectively. Traders who think conditionally spot opportunities others miss.

#What is Bayes' theorem and how does it apply to prediction markets?

Bayes' theorem is a formula for updating probabilities based on new evidence. It calculates how your belief in an outcome should change when you observe new information. In prediction markets, Bayesian updating helps traders adjust positions appropriately when news breaks—neither overreacting nor underreacting to new data.

#Can conditional probability create arbitrage opportunities?

Yes. When related markets have prices implying impossible conditional probabilities (like conditionals above 100% or below 0%), arbitrage opportunities exist. Similarly, if your calculated conditional probability based on one market differs significantly from another market's price, you may have identified a mispricing to exploit.